How to extend an Image from side to side softening the hard edge of image
You can blend the edges outwards by manipulating the texture coordinates to stretch out the image at each edge. To achieve this you can use an RGB Curves node to set the profile of the Red channel (X) to bend out the edges as follows :



To create the curve, disable clipping (click the round dot button and click the 'Use Clipping' button to disable it), create 4 additional control points, and set the 6 control points (by clicking each point in turn and typing the coordinates into the X and Y fields) to the following values :


-0.5, 0.0
0.08, 0.08
0.09, 0.09
0.91, 0.91
0.92, 0.92
1.5, 1.0


This will produce a curve that is straight in the middle section (resulting in no distortion in the middle of the image) and which flatterns at the ends (which will spread the extremeties outwards).

Note : Move points 2,3,4,5 closer to or further from the corners to blend less/more of each edge of the image. eg, moving point 2 to 0.04,0.04 and point 3 to 0.05,0.05 will blend less of the leftmost edge - the closer these points get to zero the less of the image edge will be blended. Similar for points 4 and 5 but the closer they are to 1.0. Moving points 1 and 6 closer in will reduce the extent of the blending.

Note the Mapping node to adjust the coordinates to add an additional 0.25 at the start and end of the Generated coordinate range.

This will spread the edges as shown :



This can be further manipulated using Maths nodes to 'curve' the edges with a Cosine operator and to fade out the edges as shown :



This can produce the following result :



This could potentially be enhanced by including noise in the blended section and/or the cosine to make it less uniform if desired.

Blend file attached